Premium experiment • stochastic convergence lab
STATISTICS • PROBABILITY
Monte Carlo π
Build π from deterministic pseudo-random samples, watch the estimate converge, compare two seeded runs and tune the sample size toward a target absolute error.
● LIVE RANDOM SAMPLINGπ estimate
π estimate—
Absolute error—
Reference estimate—
Std. error proxy—
Inside hits—
Target difference—
Watch probability become geometry
Uniform points fill the square while the circle-hit fraction estimates its area ratio. The seeded generator makes every experiment reproducible, so changes in error can be studied rather than dismissed as visual randomness.
Fast investigations
- Hold the seed fixed and increase the sample count by powers of two.
- Compare two seeds at the same n and observe sampling variability.
- Use the timeline to inspect early fluctuations before convergence tightens.
- Try to meet the target-error challenge with more than one seed.
LEARN WITH THE SIMULATION
Understand Monte Carlo π by experimenting.
START HERE
Monte Carlo methods estimate quantities by repeated sampling. Here, points are sampled in a square around a circle; the fraction landing inside the circle estimates the area ratio π/4.
BUILD THE IDEA
What you should understand first
- Connect random sampling to geometric probability.
- See why larger sample counts stabilize an estimate.
- Understand deterministic seeds as a way to reproduce pseudo-random experiments.
TRY IT YOURSELF
How quickly does the estimate stabilize as sample count increases?
- Use a small sample count and record the estimate.
- Increase the sample count by roughly 10×.
- Repeat again and compare how much the estimate fluctuates.
Read the visual:
The estimate usually becomes more stable with more samples, but random error decreases gradually rather than disappearing immediately.
WHY IT WORKS 01
Connect random sampling to geometric probability.
Monte Carlo methods estimate quantities by repeated sampling. Here, points are sampled in a square around a circle; the fraction landing inside the circle estimates the area ratio π/4. Focus on the relationship between the controlled variable and the measured response rather than treating the visual change as the explanation.
WHY IT WORKS 02
See why larger sample counts stabilize an estimate.
The estimate usually becomes more stable with more samples, but random error decreases gradually rather than disappearing immediately. Use an extreme or boundary setting after the baseline trial to check whether the same reasoning still holds.
DO NOT MIX THESE UP
Common misconception
Monte Carlo convergence is statistical, not monotonic. A larger sample can occasionally look worse than a smaller one. Pseudo-random sequences are generated algorithmically even though they are useful for random-style sampling.
PRACTICE CHALLENGE
How quickly does the estimate stabilize as sample count increases?
- Use a small sample count and record the estimate.
- Increase the sample count by roughly 10×.
- Repeat again and compare how much the estimate fluctuates.
Reveal the reasoning
The estimate usually becomes more stable with more samples, but random error decreases gradually rather than disappearing immediately. Monte Carlo convergence is statistical, not monotonic. A larger sample can occasionally look worse than a smaller one. Pseudo-random sequences are generated algorithmically even though they are useful for random-style sampling.
MODEL BOUNDARIES
What this simulation assumes
Every simulation simplifies reality or a mathematical system. These are the assumptions and limits to keep in mind while interpreting the results.
Model assumptions
- Sample points are treated as independent draws from a uniform distribution over the square.
- The circle is ideal and centered within that square.
- The estimate converges statistically as the number of samples increases.
Model limitations
- The browser uses pseudorandom numbers rather than a physical random source.
- Finite sample counts produce random error, so individual runs need not approach π monotonically.
CONNECT IT
Where this matters
- Numerical integration
- Risk analysis
- Simulation-based optimization
WORDS THAT MATTER
Key vocabulary
- Monte Carlo method
- Numerical method based on repeated sampling.
- Seed
- Starting value for a pseudo-random generator.
- Estimator
- Rule that uses sample data to approximate a quantity.
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